SDE with Mean-Reverting Drift model
Creates and displays SDE objects whose drift rate is expressed in
mean-reverting drift-rate form and which derive from the sdeddo
class (SDE from drift and diffusion objects).
Use sdemrd
objects to simulate of sample paths of
NVARS
state variables expressed in mean-reverting drift-rate
form, and provide a parametric alternative to the linear drift form (see sdeld
). These state variables are driven by NBROWNS
Brownian motion sources of risk over NPERIODS
consecutive observation
periods, approximating continuous-time stochastic processes with mean-reverting
drift-rate functions.
The sdemrd
object allows you to simulate any vector-valued SDEMRD
of the form:
where:
Xt is an
NVARS
-by-1
state vector of process
variables.
S is an
NVARS
-by-NVARS
matrix of mean
reversion speeds.
L is an
NVARS
-by-1
vector of mean
reversion levels.
D is an
NVARS
-by-NVARS
diagonal matrix,
where each element along the main diagonal is the corresponding element of
the state vector raised to the corresponding power of
α.
V is an
NVARS
-by-NBROWNS
instantaneous
volatility rate matrix.
dWt is an
NBROWNS
-by-1
Brownian motion
vector.
creates a SDEMRD
= sdemrd(___,Name,Value
)SDEMRD
object with additional options specified
by one or more Name,Value
pair arguments.
Name
is a property name and Value
is
its corresponding value. Name
must appear inside single
quotes (''
). You can specify several name-value pair
arguments in any order as
Name1,Value1,…,NameN,ValueN
.
The SDELD
object has the following displayed Properties:
StartTime
— Initial observation time
StartState
— Initial state at time
StartTime
Correlation
— Access function for the
Correlation
input argument, callable as a
function of time
Drift
— Composite drift-rate function,
callable as a function of time and state
Diffusion
— Composite diffusion-rate
function, callable as a function of time and state
Speed
— Access function for the input
argument Speed
, callable as a function of
time and state
Level
— Access function for the input
argument Level
, callable as a function of
time and state
Alpha
— Access function for the input
argument Alpha
, callable as a function of
time and state
Sigma
— Access function for the input
argument Sigma
, callable as a function of
time and state
Simulation
— A simulation function or
method
interpolate | Brownian interpolation of stochastic differential equations |
simulate | Simulate multivariate stochastic differential equations (SDEs) |
simByEuler | Euler simulation of stochastic differential equations (SDEs) |
When you specify the required input parameters as arrays, they are associated with a specific parametric form. By contrast, when you specify either required input parameter as a function, you can customize virtually any specification.
Accessing the output parameters with no inputs simply returns the original input specification. Thus, when you invoke these parameters with no inputs, they behave like simple properties and allow you to test the data type (double vs. function, or equivalently, static vs. dynamic) of the original input specification. This is useful for validating and designing methods.
When you invoke these parameters with inputs, they behave like functions, giving the
impression of dynamic behavior. The parameters accept the observation time
t and a state vector
Xt, and return an array of appropriate
dimension. Even if you originally specified an input as an array,
sdemrd
treats it as a static function of time and state, by that
means guaranteeing that all parameters are accessible by the same interface.
[1] Ait-Sahalia, Y. “Testing Continuous-Time Models of the Spot Interest Rate.” The Review of Financial Studies, Spring 1996, Vol. 9, No. 2, pp. 385–426.
[2] Ait-Sahalia, Y. “Transition Densities for Interest Rate and Other Nonlinear Diffusions.” The Journal of Finance, Vol. 54, No. 4, August 1999.
[3] Glasserman, P. Monte Carlo Methods in Financial Engineering. New York, Springer-Verlag, 2004.
[4] Hull, J. C. Options, Futures, and Other Derivatives, 5th ed. Englewood Cliffs, NJ: Prentice Hall, 2002.
[5] Johnson, N. L., S. Kotz, and N. Balakrishnan. Continuous Univariate Distributions. Vol. 2, 2nd ed. New York, John Wiley & Sons, 1995.
[6] Shreve, S. E. Stochastic Calculus for Finance II: Continuous-Time Models. New York: Springer-Verlag, 2004.
diffusion
| drift
| nearcorr
| sdeddo
| simByEuler